My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
January 2nd, 2014, 07:06 PM   #1
Senior Member
 
shunya's Avatar
 
Joined: Oct 2013
From: Far far away

Posts: 429
Thanks: 18

Infinite sets

Please look at the picture and answer the question.

For question 36. I tried drawing rays from the center of the circle. This does give a 1-1 correspondence between the points on the circle and the triangle outside. BUT as the rays travel outside to meet the sides of the triangle they diverge. This causes some points on the triangle to be missed out (no 1-1 correspondence).

For question 37. I tried to draw rays from a point inside the triangle. Again we get a 1-1 correspondence as required but due to divergence of the rays some points on the circle miss out.
Attached Images
File Type: png Untitled.png (22.6 KB, 80 views)
shunya is offline  
 
January 2nd, 2014, 07:53 PM   #2
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Re: Infinite sets

Can you demonstrate a one-to-one correspondence between the points of two line segments of differing lengths?
MarkFL is offline  
January 2nd, 2014, 08:19 PM   #3
Senior Member
 
shunya's Avatar
 
Joined: Oct 2013
From: Far far away

Posts: 429
Thanks: 18

Re: Infinite sets

Quote:
Originally Posted by MarkFL
Can you demonstrate a one-to-one correspondence between the points of two line segments of differing lengths?
This is my best attempt. Attempting to make a 1-1 correspondence between points on segment DE and segment BC. There is a problem in that the lines from A diverge leaving some points on BC without a partner point on DE. However, if I draw segments perpendicular to BC e.g. segment FG there is a 1-1 correspondence between the line segment BC and BA+AC.
Where am I going wrong?
Attached Images
File Type: png Untitled2.png (2.6 KB, 74 views)
shunya is offline  
January 2nd, 2014, 08:40 PM   #4
Global Moderator
 
Joined: Dec 2006

Posts: 20,931
Thanks: 2205

It's impossible to identify a point on BC without a partner point on DE.
skipjack is online now  
January 2nd, 2014, 09:17 PM   #5
Senior Member
 
shunya's Avatar
 
Joined: Oct 2013
From: Far far away

Posts: 429
Thanks: 18

Re:

Quote:
Originally Posted by skipjack
It's impossible to identify a point on BC without a partner point on DE.
What about the diverging lines? The lines from A always cut the segments DE and BC into unequal lengths. How can there be a 1-1 correspondence for ALL the points on the two segments (DE and BC)?
shunya is offline  
January 2nd, 2014, 11:22 PM   #6
Math Team
 
mathbalarka's Avatar
 
Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Infinite sets

Quote:
Originally Posted by shunya
What about the diverging lines?
The divergent lines doesn't miss anything -- it's just that neither of the set of points on both lines are countable. Use the definition of bijectivity here. For any point on BG, join it with A and intersect with DE. You have surjectivity. For injectivity, note that only a single straightline can be formed with 2 points.
mathbalarka is offline  
January 3rd, 2014, 05:03 AM   #7
Math Team
 
Joined: Sep 2007

Posts: 2,409
Thanks: 6

Re: Infinite sets

Quote:
Originally Posted by shunya
Please look at the picture and answer the question.

For question 36. I tried drawing rays from the center of the circle. This does give a 1-1 correspondence between the points on the circle and the triangle outside. BUT as the rays travel outside to meet the sides of the triangle they diverge. This causes some points on the triangle to be missed out (no 1-1 correspondence).

For question 37. I tried to draw rays from a point inside the triangle. Again we get a 1-1 correspondence as required but due to divergence of the rays some points on the circle miss out.
You are mistaking your picture for what really happens- there is a separate ray for each point. Your image of some points of the triangle being "missed out" is incorrect. For every such "missed out" point there is a ray that you have not drawn that crosses it.
HallsofIvy is offline  
January 3rd, 2014, 06:08 AM   #8
Global Moderator
 
Joined: Dec 2006

Posts: 20,931
Thanks: 2205

Quote:
Originally Posted by shunya
How can there be a 1-1 correspondence for ALL . . .
There is a 1 to 1 correspondence between each integer n and each even integer 2n, yet the set of integers seems twice as large as the set of even integers. However, the point you made is valid for finite sets.

Another example: the arctan function, defined on all the reals, is invertible, so it provides a 1-1 correspondence between all real numbers and the open interval (-?/2, ?/2).
skipjack is online now  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
infinite, sets



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Can two infinite sets be considered equal if... Giddyotwiggy Applied Math 1 June 10th, 2011 02:48 AM
onto functions..............infinite sets conradtsmith Applied Math 1 March 24th, 2010 04:07 AM
infinite independent equivalent sets scoracle Applied Math 4 October 9th, 2009 08:28 AM
Question Reguarding complements and infinite sets DoctorHandles Advanced Statistics 2 February 13th, 2009 01:25 PM
Majorities in infinite sets Infinity Applied Math 15 January 1st, 2007 04:24 PM





Copyright © 2019 My Math Forum. All rights reserved.